The topological fundamental group and free topological groups

TL;DR

This paper investigates the topological fundamental group, computes it for suspension spaces, and shows it often fails to be a topological group, linking it to free topological groups and classification problems.

Contribution

It provides explicit computations of the topological fundamental group for suspension spaces and demonstrates many cases where it is not a topological group, connecting to free topological groups.

Findings

Computed $ opgrp$ for suspension spaces $oldsymbol{}( X)$

Showed $ opgrp$ often fails to be a topological group

Linked the problem to classification issues in topology

Abstract

The topological fundamental group $\pi_{1}^{top}$ is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary space $X$, we compute the topological fundamental group of the suspension space $\Sigma(X_+)$ and find that $\pi_{1}^{top}(\Sigma(X_+))$ either fails to be a topological group or is the free topological group on the path component space of $X$. Using this computation, we provide an abundance of counterexamples to the assertion that all topological fundamental groups are topological groups. A relation to free topological groups allows us to reduce the problem of characterizing Hausdorff spaces $X$ for which $\pi_{1}^{top}(\Sigma(X_+))$ is a Hausdorff topological group to some well known classification problems in topology.